Can neural networks evolve other neural networks?
Yes, this is an active area of research as we speak. Both using classic algorithms (decision trees, random forests, Bayesian ensembles) as well as neural networks. This can also be done via evolutionary algorithms. I have personally used them for hyperparameter tuning in a few cases where squeezing out a couple of extra points of accuracy was key. 

This is in fact what Google is doing with their AutoML system. They are using neural networks for architecture search. 

Here is a Github repo with some interesting papers and links on the topic you are describing: https://github.com/hibayesian/awesome-automl-papers.